Constrained Symplectic and Contact Hamiltonian Systems: A Review

TL;DR

This review explores the geometric structures of constrained Hamiltonian systems in symplectic and contact geometries, focusing on degeneracies, constraints, and admissible phase space subsets.

Contribution

It systematically develops constraint algorithms for pre-symplectic and pre-contact manifolds, illustrating their application with examples.

Findings

Developed geometric constraint algorithms for pre-symplectic and pre-contact systems.

Clarified the structure of admissible phase space in constrained Hamiltonian dynamics.

Provided illustrative examples for each constraint algorithm.

Abstract

Singular theories, characterised by the presence of degeneracies in their Lagrangian or Hamiltonian descriptions, require the systematic implementation of constraints in order to obtain well-defined dynamics. While the symplectic framework provides the standard geometrical setting for conservative mechanical systems, those theories which exhibit dissipative effects are most appropriately discussed within the context of contact geometry. In this review, we present the geometrical structure underlying pre-symplectic and pre-contact manifolds, and develop the corresponding constraint algorithms that determine the admissible subset of phase space upon which consistent Hamiltonian evolution exists. We then close the discussion of each of the constraint algorithms with an example.